Cost Ndim Metric Metric At Equilibrium Eq Hessian

The metric at equilibrium

A Hessian matrix is the square table of second derivatives of a function, the object that describes how a surface bends near a point. For a cost function that measures the price of a recognition event, the Hessian at any point tells you the local curvature of that price. The declaration metric_at_equilibrium_eq_hessian states a compact identity: at the equilibrium point, where all coordinates are zero, the metric entry equals the corresponding Hessian entry. In symbols, for any vector α and indices i and j, the metric entry at zero is α_i times α_j, which is exactly the outer-product form of the Hessian matrix.

The identity is a theorem in the machine-checked library of formal theorems, proved by unfolding definitions and simplifying. It holds for any finite dimension n, so it is not a special-case observation. The practical content is that at equilibrium, the metric and the Hessian coincide, meaning the curvature model and the second-derivative model agree exactly. This is a bookkeeping fact: when the cost is at its zero point, the local geometry of the cost surface is fully captured by the Hessian, and no separate metric construction is needed.

What the theorem does not claim is more restrictive. It does not assert that the metric equals the Hessian away from equilibrium; the identity is anchored at the zero point only. It does not claim that the Hessian itself is derived from first principles, only that the metric entry is defined as the Hessian entry. And it says nothing about the physical meaning of the coordinates or the vector α, which remain modeling choices. The theorem is a formal bridge between two definitions, not a statement about the world.

/-- The metric at equilibrium coincides with the outer-product Hessian model. -/
theorem metric_at_equilibrium_eq_hessian {n : ℕ} (α : Vec n) :
    metricEntry α (fun _ => 0) = hessianMatrix α := by
  funext i j
  simp [hessianMatrix]

@[simp] theorem metricEntry_zero {n : ℕ} (α : Vec n) (i j : Fin n) :
    metricEntry α (fun _ => 0) i j = α i * α j := by
  have hdot : dot α (fun _ => 0) = 0 := by
    unfold dot
    simp
  simp [metricEntry, hessianEntry, hdot]

What this page does not claim

The metric equals the Hessian only at equilibrium, not at other points. The Hessian itself is not derived from first principles; it is a definitional choice. The theorem says nothing about the physical meaning of the coordinates or the vector α.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Metric.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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